Chapter 11 Angular Momentum C HAP TE R O UTL I N E 11.1 The Vector Product and Torque 11.2 Angular Momentum 11.3 Angular Momentum of a Rotating Rigid Object 11.4 Conservation of Angular Momentum 11.5 The Motion of Gyroscopes and Tops 11.6 Angular Momentum as a Fundamental Quantity 336 Mark Ruiz undergoes a rotation during a dive at the U.S. Olympic trials in June 2000. He spins at a higher rate when he curls up and grabs his ankles due to the principle of conser- vation of angular momentum, as discussed in this chapter. (Otto Greule/Allsport/Getty) T he central topic of this chapter is angular momentum, a quantity that plays a key role in rotational dynamics. In analogy to the principle of conservation of linear mo- mentum, we find that the angular momentum of a system is conserved if no external torques act on the system. Like the law of conservation of linear momentum, the law of conservation of angular momentum is a fundamental law of physics, equally valid for relativistic and quantum systems. 11.1 The Vector Product and Torque An important consideration in defining angular momentum is the process of multiply- ing two vectors by means of the operation called the vector product . We will introduce the vector product by considering torque as introduced in the preceding chapter. Consider a force F acting on a rigid object at the vector position r (Fig. 11.1). As we saw in Section 10.6, the magnitude of the torque due to this force relative to the origin is r F sin , where is the angle between r and F . The axis about which F tends to pro- duce rotation is perpendicular to the plane formed by r and F . The torque vector is related to the two vectors r and F . We can establish a mathe- matical relationship between , r , and F using a mathematical operation called the vector product, or cross product: r F (11.1) We now give a formal definition of the vector product. Given any two vectors A and B , the vector product A B is defined as a third vector C , which has a magnitude of AB sin , where is the angle between A and B . That is, if C is given by C A B (11.2) then its magnitude is C AB sin (11.3) The quantity AB sin is equal to the area of the parallelogram formed by A and B , as shown in Figure 11.2. The direction of C is perpendicular to the plane formed by A and B , and the best way to determine this direction is to use the right-hand rule illustrated in Figure 11.2. The four fingers of the right hand are pointed along A and then “wrapped” into B through the angle . The direction of the upright thumb is the direc- tion of A B C . Because of the notation, A B is often read “ A cross B ”; hence, the term cross product. Some properties of the vector product that follow from its definition are as follows: 1. Unlike the scalar product, the vector product is not commutative. Instead, the or- der in which the two vectors are multiplied in a cross product is important: A B B A (11.4) 337 O r P φ x F y τ = r × F z τ Active Figure 11.1 The torque...
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